SOLUTION: Find the center and radius of a circle circumscribing a triangle with the vertices (5,1) (-6,0), (-1,-7)

Question 937517: Find the center and radius of a circle circumscribing a triangle with the vertices (5,1) (-6,0), (-1,-7)
Answer by TimothyLamb(4379) (Show Source): You can put this solution on YOUR website!
A(5,1)
B(-6,0)
C(-1,-7)
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find slope for chord AB:
mab = (1-0)/(5+6)
mab = 1/11
---
find midpoint of chord AB:
x = (5-6)/2
y = (1+0)/2
x = -1/2
y = 1/2
---
find slope for chord BC:
mbc = (0+7)/(-6+1)
mbc = -7/5
---
find midpoint of chord BC:
x = (-6-1)/2
y = (0-7)/2
x = -7/2
y = -7/2
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the center of the circle is at the intersection of the two lines that pass through the midpoints of each chord and that are perpendicular to each chord
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find the equation for line perpendicular to chord AB:
slope = -1/mab = -11
y - 1/2 = -11(x + 1/2)
y = -11x - 11/2 + 1/2
y = -11x - 5
---
find the equation for line perpendicular to chord BC:
slope = -1/mbc = 5/7
y + 7/2 = (5/7)(x + 7/2)
y + 7/2 = (5/7)x + (5/7)(7/2)
y + 7/2 = (5/7)x + 35/14
y = (5/7)x + 35/14 - 7/2
y = (5/7)x + 35/14 - 49/14
y = (5/7)x - 1
---
find the center of the circle ...
by finding the intersection of the linear system:
y = -11x - 5
y = (5/7)x - 1
---
put the system of linear equations into standard form
---
-11x - y = 5
0.714285714286x - y = 1
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---
solution, center of the circle:
x = -0.34146341
y = -1.2439024
---
find the radius of the circle ...
the distance from the center to one of the points on the circle ...
using point A:
---
r = sqrt( (-0.34146341 - 5)^2 + (-1.2439024 - 1)^2 )
r = 5.7936456
---
using point B:
---
r = sqrt( (-0.34146341 - -6)^2 + (-1.2439024 - 0)^2 )
r = 5.7936456
---
check ...
the general equation for a circle:
---
(x - a)^2 + (y - b)^2 = r^2
(x + 0.34146341)^2 + (y + 1.2439024)^2 = 5.7936456^2
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at point A:
(5 + 0.34146341)^2 + (1 + 1.2439024)^2 = 33.566329
33.566329 = 33.566329
---
at point B:
(-6 + 0.34146341)^2 + (0 + 1.2439024)^2 = 33.566329
33.566329 = 33.566329
---
at point C:
(-1 + 0.34146341)^2 + (-7 + 1.2439024)^2 = 33.566329
33.566330 = 33.566329
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